On joint estimates for maximal functions and singular integrals on weighted spaces
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- by Maria Carmen Reguera and James Scurry PDF
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Abstract:
We consider a conjecture attributed to Muckenhoupt and Wheeden which suggests a positive relationship between the continuity of the Hardy-Littlewood maximal operator and the Hilbert transform in the weighted setting. Although continuity of the two operators is equivalent for $A_p$ weights with $1 < p < \infty$, through examples we illustrate this is not the case in more general contexts. In particular, we study weights for which the maximal operator is bounded on the corresponding $L^p$ spaces while the Hilbert transform is not. We focus on weights which take the value zero on sets of nonzero measure and exploit this lack of strict positivity in our constructions. These types of weights and techniques have been explored previously by the first author and independently with C. Thiele.References
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Additional Information
- Maria Carmen Reguera
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- Address at time of publication: Centre for Mathematical Sciences, University of Lund, Lund, Sweden
- Email: mreguera@math.gatech.edu, mreguera@maths.lth.se
- James Scurry
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- Email: jscurry3@math.gatech.edu
- Received by editor(s): September 9, 2011
- Published electronically: November 19, 2012
- Additional Notes: The first author’s research was supported in part by grant NSF-DMS 0968499
The second author’s research was supported in part by the National Science Foundation under grant No. 1001098 - Communicated by: Michael T. Lacey
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1705-1717
- MSC (2010): Primary 42B20; Secondary 42B25, 42B35
- DOI: https://doi.org/10.1090/S0002-9939-2012-11474-1
- MathSciNet review: 3020857